Zero total flux for localized tube Stokes flow ($Q_f(z)=-V_z{A}_s(z)$)

Extend the velocity rigidly through a moving sphere in an incompressible tube flow. The extended velocity is continuous and divergence-free, so its total flux is independent of height. Quiescent far-field conditions make this flux zero. The sphere contributes $V_z{A}_s$ through a cross-section; the rotational contribution integrates to zero over its circular slice. Thus the fluid flux compensates the solid displacement. A descending sphere generates upward fluid flux in every section cutting its interior. This statement assumes a localized flow with no imposed throughflow and negligible tube-end effects.